Paul Cohen
Paul Cohen
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Full Name and Common Aliases
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Paul Joseph Cohen was a mathematician whose full name is often cited in academic circles.
Birth and Death Dates
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Paul Cohen was born on April 2, 1934, and passed away on March 23, 2007.
Nationality and Profession(s)
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Cohen held American citizenship and was primarily known as a mathematician. He worked extensively in the field of set theory, particularly with regards to the Continuum Hypothesis and forcing.
Early Life and Background
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Paul Cohen was born in Long Branch, New Jersey, USA. Growing up, he showed an early aptitude for mathematics and was encouraged by his parents to pursue this interest. He attended Stuyvesant High School in New York City before enrolling at the University of Chicago, where he earned both his Bachelor's (1954) and Master's degrees (1955). Cohen then moved to Stanford University to work on his Ph.D., which he completed in 1960 under the supervision of Alfred Tarski.
Major Accomplishments
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Cohen is best known for resolving a significant problem in set theory. In 1963, he proved that the Continuum Hypothesis (CH) is independent of Zermelo-Fraenkel axioms with or without the Axiom of Choice. This involved developing his forcing method, which revolutionized mathematical logic and had far-reaching implications for various areas of mathematics.
Notable Works or Actions
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Cohen's work on set theory not only resolved a long-standing problem but also opened up new avenues for research in this field. His contributions to the development of modern set theory have been widely acknowledged, with many considering him one of the most important logicians of the 20th century.
Impact and Legacy
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Cohen's resolution of the Continuum Hypothesis is considered a landmark moment in mathematics. It has had lasting impacts on the study of mathematical logic, particularly within set theory. Cohen's work continues to be studied by mathematicians today, with its influence extending beyond the field of pure mathematics into other disciplines such as philosophy and computer science.
Why They Are Widely Quoted or Remembered
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Paul Cohen is widely recognized for his groundbreaking contributions to set theory. His proof of the independence of the Continuum Hypothesis has had a lasting impact on mathematical logic, making him one of the most influential mathematicians of the 20th century. This is why he continues to be quoted and remembered as a pioneer in the field.
Quotes by Paul Cohen
If I buy a hardcover biology textbook today, it was written three years ago. With an online textbook, the publisher can change it immediately, so the kids are getting up-to-date information. That's a tremendous difference.
They capitalized on some mistakes we made, ... It was an exciting game, but we fell short.
The numbers Visa has seen over the Friday and Saturday period are encouraging, but we will continue to watch over the next several weeks to see if the positive trends hold true, ... Based on the data Visa has seen to date, we remain cautiously optimistic that this will be a good holiday season for merchants.
The odd thing is that the initial rush in some stores has dissipated, ... I think the shopping rush could taper off over the weekend.
It's an interesting dichotomy, ... We know we're the defending state champ, so we've got to be ready for that.
It might make sense for it to be part of Ayer, because with Devens you'd have another community with no access to it.
There's no question we're going to be short on those accounts. But when you're dealing with a 15,16, or 17 million dollar budget, you keep things in perspective... we'll have to swallow it and move forward.
There's no question we're the underdog again, Brady has very good skill position players. But we've been in this position before and we've risen to the challenge.
Black Friday is like a national phenomenon, ... The nice thing this year is that there's a lot of mall store traffic very early and people are holding lots of bags. So they're not just window shopping. They're actually buying.
When all the numbers are counted, consumers could have given retailers an early holiday present, ... Last year Black Friday spending on our cards totaled $3.6 billion. We could cross $4 billion this year.